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The Call Of The Primes

Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics

Prometheus Books

SOME WORDS ON THE LO SHU AND OTHER MAGIC SQUARES

Magic squares have fascinated people of all ages through the centuries. A magic square is an array of numbers, usually distinct, arranged in square formation. Thus a magic square contains the same number of horizontal rows as vertical columns. The magic square derives its name from the fact that the sums of the numbers in each row, in each column, and across its two diagonals, are identical. Although no mathematical knowledge is imparted through the study of magic squares, nevertheless many have investigated these squares with the expectation of finding beautiful relationships between the integers within the squares. Those inquisitive investigators — many of whom were amateur mathematicians — have found many beautiful properties! Even great professional mathematicians through the ages have inquired into magic squares, including one of the greatest mathematicians of all time: Leonard Euler.

A normal magic square is one in which the integers from 1 to n are arranged in a square pattern so that each row, each column, and each of the two diagonals add up to a specific integer. It is this property that makes the magic square magical. The earliest magic square is said to date back to 2800 BCE in China. According to a famous Chinese myth, while walking beside the Lo River, Emperor Yu found a tortoise with a specific pattern on its shell. He named this pattern the lo shu. (The word lo is the name of the river and the word shu means books.) The lo shu consisted of a number of dots representing whole numbers that added up to 15 in three horizontal directions, three vertical directions, and two diagonal directions.

Although the belief prevails to this day that magic squares were first discovered well over four thousand years ago, present-day Chinese scholars can only trace its earliest discovery back as far as the fourth century BCE. It is believed today that magic squares migrated from China to India around the fifth century CE and they later spread to Arab cultures. The Arabs used them in practicing astrology and other forms of magic rituals. To this day magic squares are still used in various parts of the world as amulets or good luck charms.

The ancient Chinese attached great significance to the three-by-three magic square. They saw the even numbers as representing yin, the female principle, and the odd numbers as representing yang, the male principle. The central number 5 represented the earth; 4 and 9 represented metal; 2 and 7 represented fire; 6 and 1, water; and 8 and 3, wood. Thus the four elements were found to exist in the three-by-three square.

The lo shu today is usually written as follows:

[ILLUSTRATION OMITTED]

The number of rows or columns of a magic square is said to be its order. Thus the three-by-three magic square shown in figure 1.1 is said to be of the order three. Its three rows, three columns, and each of its two diagonals add up to 15, which is said to be the magic square's constant. Of course, it is possible to rotate or reflect the lo shu. You could, for example, rotate the above magic square such that the right column becomes the top row. The top horizontal row would then be 2, 7, 6, or it could be its reflection (i.e., its reversal), 6, 7, 2; therefore, the middle row could contain the digits 9, 5, 1 or 1, 5, 9, respectively, and the bottom horizontal row could contain the digits 4, 3, 8 or 8, 3, 4. These reflections and rotations, however, do not change the essential nature of the three-bythree magic square. Thus the order-three magic square is unique.

No order-one or order-two magic squares are possible. An order-one magic square can only contain one number and is therefore considered trivial. By convention, mathematicians agree that an order-one magic square does not exist.

Suppose a magic square of order two exists. Substituting letters for numbers, we would have in our square:

AB

CD

Since these four numbers constitute a magic square (where all four digits are distinct) we know that

A + B = A + C.

Subtracting A from both sides of this equation gives

B = C.

Therefore, B = C. However all the numbers in a magic square must be distinct. Since our magic square of order two must contain at least two similar numbers, we conclude that an order-two magic square cannot exist.

All order-three magic squares are of the form shown in figure 1.2:

[ILLUSTRATION OMITTED]

Here, a, b, and c are positive integers such that a is less than b and b is less than c - a, and b does not equal 2a. To obtain the smallest possible order-three magic square, let a = 1; b = 3, and c = 5. Plugging these values in to the magic square shown in figure 2.2, the top horizontal row contains the digits — from left to right — 4, 9, 2; the digits in the middle row are 3, 5, 7; and the digits in the bottom row are 8, 1, 6.

The constant of a normal magic square is easily derived. The sum of the integers from 1 to n is (n (n + 1))/ 2, or (n2 + n)/ 2. The integers that appear in a normal magic square run from 1 to n2. Thus the sum of the integers in a standard magic square is n2 (n2 + 1)/2 or (n4 + n2)/2. The magic square consists of n rows. Therefore, the sum of each row is (n + n)/2n, which equals (n3 + n)/2. Thus the constant of a normal magic square is (n3 + n)/2. Hence the constant of a normal three-by-three magic square is (33 + 3)/2, or 15. The constant of a normal four-by-four magic square is 34; the constant of a normal five-by-five magic square is 65. The series continues with 111, 175, 260, 369, 505, 671, and so on.

Curiously, the lo shu magic square just barely exists. In the lo shu, there are three rows, three columns, and two diagonals, which each sum to 15. Thus for a normal order-three magic square to exist, eight subsets consisting of triplets of numbers whose sum is 15 must exist in the range of digits from 1 to 9. Fortunately, exactly eight such triplets exist. These are: 1,5,9; 2,9,4; 2,5,8; 2,7,6; 3,5,7; 4,3,8; 8,1,6; 6,5,4.

A magic square that includes zero can exist. A unique three-by-three magic square that has a constant of 15 is possible if a zero is included, and if 1 and 9 are excluded. In this case, the three numbers — including zero — that add to 15 must consist of 0, 7, and 8; or 0, 5, and 10. There is no other way that three numbers — if one of them is zero — summed together will equal 15. Therefore, 0 cannot occupy a corner cell, because if it did, it would have to be a part of three triples of numbers that add to 15, and this impossible. Since 5 must occupy the center cell, we find that this magic square must have 10 above the 5 and 0 below it. The numbers 7 and 8 will be on either side of the 0. Thus the top row will consist of 2, 10, and 3; the middle row will contain 6, 5, and 4; and the bottom row will consist of 7, 0, and 8. Of course, an infinite number of three-by-three magic squares that contain multiples of these numbers is possible.

Magic squares of order n, commencing with integer A, may be formed where the numbers within them are increasing in an arithmetic series with a difference of D between terms. The constant of such a magic square may be obtained by using the following simple formula:

1/2 n(2A + (D (n2 - 1))).

Thus in the simplest three-by-three magic square the smallest integer is 1, and the integers are increasing in an arithmetic series, with a common difference of 1 between terms. Thus, in the above expression A = 1, D = 1, and n = 3. The expression then produces a constant that equals

1/2 • 3 • (2 • 1 + (1 • (32 - 1))).

This equals 1/2 • 3 • (2 + 8), which equals 15. Therefore, the constant of the simplest three-by-three magic square is 15.

Suppose you begin to construct a three-by-three magic square where the smallest term equals 1, and where the terms are increasing in an arithmetic series, with a common difference of 2 between terms. Thus, in such a square A equals 1, D equals 2, and n equals 3. The above expression then produces a constant that equals

1/2 • 3 • (2 • 1 + (2• (32 - 1))).

This expression equals 27. Thus the constant of a three-by-three magic square, where the smallest integer is 1, and the terms are increasing in an arithmetic sequence with a difference of 2, is 27. Figure 1.3 shows this particular three-by-three magic square.

An existing magic square may be used to create a new one. Subtract every number in an order n magic square from n2 + 1, and a new square, called the complementary of the first square, is formed.

For example, suppose one forms a three-by three magic square by multiplying each term in the lo shu by 30. This creates the following three-by-three magic square.

The constant in the three-by-three magic square shown in figure 1.4 is 450, and the value of n is, of course, 3. Therefore, n2 - 1 equals 8. If one subtracts each term in the square in figure 1.4 from 8, one obtains the three-by-three square shown in figure 1.5.

Of course, in summing the negative terms in the magic square shown in figure 1.5, you may ignore the negative sign in front of each integer, and you may sum the integers in the magic square as if all the terms are positive integers. You will then find that the constant in this magic square is 426.

If any specific integer is added to or subtracted from each of the numbers in a magic square, a new magic square is formed. The same holds if each of the integers in a magic square is multiplied by a constant specific integer. For instance, consider the four-by-four magic square shown in figure 1.6. The constant in this square is 34. If one adds, say, 12 to each term, the result is the magic square as shown in figure 1.7, which has a constant of 82. On the other hand, if each term in the magic square shown in figure 1.6 is multiplied by some constant, say 23, the result is the magic square that is shown in figure 1.8, which has a constant of 782.

No one yet has discovered a universal formula that determines the numbers of magic squares of a given order, n. It is known that there is just one magic square of order three (ignoring reflections and rotations). There are 880 order-four magic squares, ignoring reflections, rotations, and such like. There are 275,305,224 order-five magic squares. Beyond order five the exact number of magic squares of order n is unknown, although various estimates for n up to the value of 12 have been formulated and published.

Many beautiful properties unique to the lo shu have been discovered. There are almost certainly many more waiting to be unearthed. I will mention just a few that have been found. The sum of the squares of the integers in the top row is equal to the sum of the squares in the bottom row. The same property holds for the far-left and far-right columns. Consider the digits in each of the three rows as three-digit numbers, reading from left to right. These three-digit numbers are 492, 357, and 816. They sum to 1,665. So also do their reversals: 618, 753, and 294. A similar property applies to the numbers in the columns. It is also curious that 49 - 2 + 35 - 7 = 81 - 6.

The beautiful order embedded in the lo shu also reveals itself when you associate it with the prime numbers, those enchanting integers that are the building blocks of all composites. (A composite number is an integer, greater than 1, that is not a prime number.) Here are a couple of examples of the lo shu's relationship with the primes. Reading the columns in the lo shu from the bottom up, the three three-digit numbers 834, 159, and 672 appear. The 834th prime is 6,397; the 159th prime is 937; and the 672nd prime is 5,011. The sum of 6,397, 937, and 5,011 is 12,345, a number consisting of five consecutive digits. Reading the columns from the top down, you encounter the three-digit numbers 438, 951, and 276. The 438th prime is 3,061; the 951st prime is 7,507; and the 276th prime is 1,783. The sum of 3,061, 7,507, and 1,783 is 12,345 plus 6.

Reading across the rows of the lo shu from right to left, the following three three-digit numbers appear: 294, 753, and 618. Those three numbers, as well as their reversals, add up to 1,665. The same applies to the three threedigit numbers found in the vertical columns, 438, 951, and 276 (and their reversals). The number of primes less than 1,665 is 261. There are nine cells in the lo shu. Curiously, 261 equals 9 times the ninth odd prime. Also, 261 = 4 - 92 + 357 + 8 - 16.

Magic squares of various orders often display beautiful patterns when lines are drawn connecting consecutive integers. When such lines are drawn in the lo shu, for example, the following pattern emerges:

[ILLUSTRATION OMITTED]

In 1984, Dr. Martin LaBar, a professor of science at Southern Wesleyan University, in Central, South Carolina, asked an apparently simple question: Does a magic square of order three exist that contains nine distinct square integers? The late Martin Gardner, the famous writer on recreational mathematics who was mentioned in the introduction, posed the same problem in his column in the magazine Quantum in 1996. Gardner remarked in that same column that if a three-by-three magic square of squares exists, which contains nine distinct entries, each of the square numbers within such a square "are sure to be monstrously large"

As a consequence of Gardner's article, it is believed that many professional and amateur mathematicians, and a wide assortment of computer programmers around the world, began tackling the puzzle. However, the problem is still unsolved to this day. In 1997, Lee Sallows, a computer programmer in the Netherlands and a widely renowned expert on magic squares, discovered a close miss when he found the three-by-three square shown in figure 1.10, where the nine entries are all perfect squares.

Three rows, three columns, and one diagonal of the square shown in this figure all sum to the same constant: 21,609. Unfortunately, the second diagonal sums to 38,307. Sallows published his result in an article in the Mathematical Intelligencer in 1997.

Gardner's insight into the difficulty of the problem was confirmed in 1998 when Duncan Buell, of the Department of Computer Science and Engineering at the University of South Carolina, computed that the central cell of a threeby-three magic square of squares would have to be greater than 25 times 10 raised to the twenty-fourth power (24 • 1024).

Magic squares of order four are believed to have been first discovered in India around 1000 CE. The constant of a normal four-by-four magic square is (43 + 4)/2, or 34. Excluding rotations and reflections, the number of fourby-four magic squares is 880. The four-by-four square, shown in figure 1.11, appears in a famous 1514 engraving by Albrecht Durer (1471-1528), the German engraver and mathematician. The engraving is titled Melencolia I.

The constant of Durer's square is 34. There are many beautiful properties in the square, but I will mention just a few. The numbers in each of the four quadrants sum to 34: 16, 3, 10, and 5; 2, 13, 8, and 11; 7, 12, 1, and 14; and 9, 6, 15, and 4. The sum of the numbers in the four corners, 16, 13, 4, and 1, sum to 34. The corner numbers in each of the four three-by-three grids within Durer's square also sum to 34. The four numbers in the central square, 10, 11, 7, and 6, sum to 34. The two central numbers in the top row, 3 and 2, added to the two central numbers in the bottom row, 15 and 14, sum to 34. The same property applies to the two central numbers in the far-left and far-right columns. The two central numbers in the bottom row of Dürer's square give the year in which the engraving was made: 1514. The numbers in the two bottom corners, 1 and 4, give the numeric positions of the initials of Albrecht Dürer.

An order-three magic square consisting of powers of 3, 4, and 5 have been known to be impossible since 1900. Can an order-three magic square consisting of powers greater than 5 exist? The answer is no.

Can an order-four magic square exist in which all of its integers are square numbers? The answer is yes. The first known magic square of any order, consisting of all square numbers, was created by the great Leonard Euler and sent to Joseph-Louis Lagrange in 1770.

Lagrange was born in Turin, Italy, in 1736. He was one of eleven children, of which only two survived into adulthood. Lagrange was mainly a self-taught mathematician, but he was brilliant at the subject. He was appointed professor of mathematics at the Royal Artillery College in Turin when he was only nineteen years old. Lagrange died in Paris, France, in 1813.

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Excerpted from The Call Of The Primes by Owen O'Shea. Copyright © 2016 Owen O'Shea. Excerpted by permission of Prometheus Books.
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